Problem: Emily is 3 times as old as Michael. Twenty years ago, Emily was 7 times as old as Michael. How old is Emily now?
We can use the given information to write down two equations that describe the ages of Emily and Michael. Let Emily's current age be $e$ and Michael's current age be $m$ The information in the first sentence can be expressed in the following equation: $e = 3m$ Twenty years ago, Emily was $e - 20$ years old, and Michael was $m - 20$ years old. The information in the second sentence can be expressed in the following equation: $e - 20 = 7(m - 20)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $e$ , it might be easiest to solve our first equation for $m$ and substitute it into our second equation. Solving our first equation for $m$ , we get: $m = e / 3$ . Substituting this into our second equation, we get: $e - 20 = 7($ $(e / 3)$ $- 20)$ which combines the information about $e$ from both of our original equations. Simplifying the right side of this equation, we get: $e - 20 = \dfrac{7}{3} e - 140$ Solving for $e$ , we get: $\dfrac{4}{3} e = 120$ $e = \dfrac{3}{4} \cdot 120 = 90$.